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STRESS CONCENTRATION AT NOTCHES One of the fundamental issues of designing a fatigue resistant structure (“design against fatigue”) is the consideration.

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Presentation on theme: "STRESS CONCENTRATION AT NOTCHES One of the fundamental issues of designing a fatigue resistant structure (“design against fatigue”) is the consideration."— Presentation transcript:

1 STRESS CONCENTRATION AT NOTCHES One of the fundamental issues of designing a fatigue resistant structure (“design against fatigue”) is the consideration of stress concentration Stress concentration at geometrical notches are always present in a real structure Notches introduce inhomogeneous stress distribution with a stress concentration at the root of the notch Stress concentration factor: K t describes the severity of the notch and depends on the geometry of the notch configuration (shape factor of the notch) K t is referred as the theoretical stress concentration factor: it is based in the assumption of linear elastic material behavior

2 Common examples of stress concentration (a)Gear teeth (b)Shaft keyway (c)Bolt threads (d)Shaft shoulder (e)Riveted or bolted joint (f)Welded joint all these components might be subjected to cyclic loads !

3 DEFINITIONS: For the previous example: therefore: following the definitions of R.E. Peterson in Stress Concentration Factors, John Wiley & Sons, New York (1974) In general K t is the preferred factor to indicate stress concentration usualalternative

4 THE “MODEL” STRESS CONCENTRATION CASE: the circular hole in an infinite sheet S S r  r0r0  for   = 0 ? Along the edge of the circular hole: compression at  = 0  = 30° or  =  /2 –  from    max

5 Stress Profiles along the normal to the edge of the circular hole: We are interested in evaluating: a) Situation for compressive remote stress (–S): presence of tensile stress ?  YES b) Gradient of Stress in the direction normal to the edge of the hole at the location of s peak : strong gradient c) Gradient of Stress along the edge of the hole at the location of s peak : slow decrease of stress along the edge of the notch d) Volume of material subjected to high Stress around the root of the notch: larger for larger notches! (significant to understand notch size effects on fatigue)

6 a)Presence of Local Tensile Stress upon Remote Compression Stress * Fatigue Crack Growth under Compressive Stress? * Effect on Brittle Materials: Failure Criteria Based on Maximum Normal Stress for Brittle Materials How would a cylinder of brittle material with a distribution of defects fail under compression? Spherical voids and sharp like cracks are often produced during processing of brittle materials: - Spherical voids  sintering of ceramic powders (remnants of initial porosity) - Microcraks  thermal expansion mismatch CC CC

7 b) Gradient of Stress in the direction NORMAL to the edge of the hole at the location of  peak * Although the peak stress is of great importance, it is also interesting to know how fast the stress decreases away from the root of the notch  Stress Gradient for the circular hole: The Stress Gradient at the root of a notch should give an indication of the volume of material under high stresses  estimate the distance  along the normal to the root for a drop from  peak to 0.9  peak (10 % decrease) for a circular hole with r 0 = 2.5 mm:  = 0.1 mm= 100  m if grain size 50  m  the depth  corresponds only to few grains The grains at the noth root surface are subjected to high loads and this is very important for fatigue

8 S S c) y d) Gradient of Stress ALONG the edge of the hole at the location of  peak As fatigue crack nucleation is a surface phenomenon it is of interest to know how fast the tangential stress along the edge of the notch is decreasing * Slow decrease of the stress along the edge compared with the decrease from the edge at the location of  peak * Larger notches have a larger material surface along the root of the notch  very important to understand notch size effects on fatigue Note: even when a particular case was analyzed here (circular hole in an infinite plate) the conclusions are of general validity  similar peak stresses and notch root radii give comparable stress distribution around the root of an arbitrary notch

9 Geometrically similar specimens have the same K t (K t is dimensionless) … Larger specimens have larger volumes and larger notch surface areas of highly stressed material Reason of the existence of Notch Size Effects in Fatigue … but different stress gradients (stress gradient is not dimensionless) Also: Importance of surface quality (method of production / fabrication)  surface defects due to manufacturing in a highly stressed region along the wall of a hole Effect of notch geometry on K t Further examples / further aspects of stress concentrators 

10 The elliptical hole in an infinite sheet a / b1/313  / a 911/9 KtKt S S use large radii on surface parallel to applied stress to reduce stress concentration !

11 Stress Concentration for an elliptical hole under biaxial loading: * For the case of a thin walled pressure vessel under pressure  = 0.5 and for the case of a circular hole (a = b): lower than 3 S for uniaxial loading * Same case but elliptical hole with b/a = 2: lower than 3 S for uniaxial loading (actually,   = 1.5 S along the edge of the hole) compare with the square hole (dashed line) with rounded corners with r 10% of hole width: K t = 4.04 for  = 0.5  : biaxiality ratio S

12 Stress Concentration for a circular hole in a plate under pure shear: Fatigue cracks growing from holes in a shaft subjected to cyclic torsion !

13 Pin - loaded hole: Comparison of K t values for a lug and an open hole * Lugs are fatigue critical parts (also prone to fretting corrosion)  values of d/W below 1/3 are usually avoided to keep K t below 3.5 Conection between a lug and a clevis:

14 Superposition of notches: If a relativelly small notch is added to the root of the main notch  Effect of superposition of notches: This overestimates K t because the small notch is not completely embedded in an homogeneous stress field of magnitude K t1

15 Technique for estimating conservative limiting value for K t for superposed notches: The theoretical stress concentration factor for the single deep narrow notch will always be greater than the K t for the multiple notch (see K t for Edge Notches two transparencies later) “Fill” the notch (cross hatched area) leaving a single deep narrow notch

16 Examples of Superposition of notches: Cross section of a fatigue crack at a sharp corner Lug with small lubrication hole to the lug hole

17 Edge notches and Corrosion Pits Corrosion pits at the material surface of an Al-alloy. Pit depth = 0.15 mm. Equivalent shape gives very high K t values

18 Stress concentration factors for a shaft with a grove subjected to: Axial Load Torsion Bending Further information of the type that can be found in the clasical handbook of R.E. Peterson, Stress Concentration Factors, 1974

19 S a = S e S eK THE FATIGUE STRENGHT OF NOTCHED SPECIMENS * STRESS – LIFE APPROACH: Notch Effects on the Fatigue Limit (S m = 0) Similarity Principle: if S a = S e is the fatigue limit of the smooth specimen, then S peak should give the fatigue limit S eK of the notched specimen: meaning that: … but this is not the case !  The Fatigue Strength Reduction Factor or Fatigue Notch Factor K f is introduced: In general: K f < K t  fatigue limit of different materials are less notch sensitive to fatigue than predicted by K t

20 Effect of a Notch on S - N behavior (Tryon and Day, 2003) Examples:

21 Mechanical Behaviour of Materials (Dowling, 1999) The examples illustrates a general observation for different materials: the finite life region is also less notch sensitive to fatigue than predicted by K t

22 In general K f < K t Blunting effects in soft materials: Yielding at the notch root reduces peak stress from the values predicted by K t Fatigue strength of a notched component depends on the volume of highly stressed material near the notch  also effect of stress gradient on crack growth. K t depends on geometry and mode of loading K f also depends on material and notch size and

23 For engineering applications, the fatigue strength reduction fator K f can be empirically related to the elastic stress concentration facto K t by a Notch Sensitivity Factor defined as: q = 1  material fully notch sensitive: K f = K t q = 0  material not notch sensitive: K f = 1 Empirical equations for q were proposed by different authors: * Peterson (1959) * Neuber (1946) * Siebel and Stiele (1955)

24 * Peterson assumed that fatigue damage occurs when a the stress at a point located at a critical distance a p away from the notch root is equal to the fatigue strength of a smooth specimen and obtained the following empirical equation: - r is the notch root radius - a P is a material constant related with material strength and ductility. Effect of notch root radius on K f * for high strenght steels with S U > 560 MPa:

25 Peterson ´s notch sensitivity for steels Also, q can be obtained in graphical form:

26 * Neuber assumed that fatigue failure occurs if the average stress over a length a N from the notch root is equal to the fatigue limit of a smooth specimen and proposed the following empirical equation: - r is the notch root radius - a N is the Neuber´s material constant related to the grain size Neuber´s Notch Sensitivity curves for Al alloys Relation with Hall-Petch?

27 where: for the circular hole: * Siebel and Stiele (1955) introduced the Relative Stress Gradient (RSG)  to characterize the effects of fatigue strength reduction (instead of using the notch radius!) No significant effect of K t on   Similar dependencies are found for other geometries and for typical K t values: 2 < K t <5

28 Testing the Fatigue Strength of smooth and notched specimens they generated empirical curves relating K f / K t vs.  Equation for the Curves: where C ss is a material constant related with S y

29 RSG (  ) values calculated from Siebel and Stiele by the theory of elasticity for different notched members Examples:

30 * STRESS – LIFE APPROACH: Notch Effect on Finite Life (S–N curve) In the Finite Life region the notch may be less sensitive to what is predicted by K f for the Fatigue Limit  to much conservative ! Use K f for the whole S-N curve? : define a Fatigue Sensitivity Factor at a particular Life (and interpolate / extrapolate) analogously to what was made for estimating S – N curves for smooth specimens  K f for N 1000 : K’ f

31 An Empirical Fatigue Notch Sensitivity Factor (q’ 1000 ) can be defined at 1000 cycles:

32 * Estimate of Fatigue Life for Notched Component: different approaches Fatigue Notch Sensitivity Factor K f depends on Cycles to Failure N f A) “Juvinall” approach B) Examples: LCF REGION!

33 Example: Juvinall for the following loading cases: (1)Reversed bending loading (2)Reversed axial loading (negligible bending) (3) Reverse torsional loading

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